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Michael Atiyah - Russian contributions to the Dirac operator (44/93)

1,601 views · Published 20 June 2017 · 1:46 · Indexed 29 October 2026

Channel: Web of Stories - Life Stories of Remarkable People · 2017 · Science & Technology

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British mathematician Michael Atiyah (1929-2019) studied in Cambridge where he became a Fellow of Trinity College and later held professorships at Princeton and Oxford. He is best known for his work on the K-theory and the Atiyah-Singer Index Theorem. [Listener: Nigel Hitchin; date recorded: 1997]

TRANSCRIPT: [NH] I had heard that one of the Russian formulas had a missing n-factorial or something because… is that right?

Well the Russians, you see, were only… they tended to consider the simple cases. They weren't concerned with the general cases. You just started off with low dimensional examples and they didn't worry with non-trivial tangent bundles, so they just had a formula involving the degree; which is the top formula that's showing character formula, and that wouldn't… the factorial, they would miss it. I mean it wouldn't be there in the first instant, but you don't put it in until you know the… but they might have thought there was some constant and wouldn't know what it was. They would know that the topological information would be contained in this degree, because in the simpler case they considered, that was the only topological information there was. And then they would guess that this was the answer, or there might be a factor, but they wouldn't have known the formula to expect, certainly, not in general, and they started off in dimensions two or three where the factorials aren't that big.

So the Russians formulated the problem, they hadn’t got… and they'd started working on the analysis of pseudo-differential operators, deforming differential operators through pseudo-differential operators so that you then begin to apply the continuity argument to say it had to depend on some symbol… so they got the general framework. But what they were totally missing was the machinery of the general algebraic topology side, which you had to plug into it. Now we had the… we came from the other end completely, in some sense we had to learn our analysis, but it was easier for us to learn the analysis than for them to learn all the elaborate machinery which I suppose was what Hirzebruch had contributed. So it was having the right friends at the right time that did it.

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